J. agric. Engng Res. (2001) 78 (4), 377d389 doi:10.1006/jaer.2000.0663, available online at http://www.idealibrary.com on PM*Power and Machinery
Optimizing Design Parameters of a Mouldboard Plough D. S. Shrestha; G. Singh; G. Gebresenbet Davidson 139, Iowa State University, Ames, IA-50010, USA; e-mail of corresponding author:
[email protected] Asian Institute of Technology, P.O. Box 4; Klong Luang, Pathumthani 12120, Thailand; e-mail:
[email protected] Swedish University of Agricultural Sciences, P.O. Box 7033, 750 07 Uppsala, Sweden; e-mail:
[email protected] (Received 19 October 1999; accepted in revised form 18 October 2000; published online 19 January 2001)
Owing to the complexity in designing a mouldboard plough, it has been manufactured mostly based on experience and empirical methods. In this research, a mathematical model of a mouldboard plough has been developed to design it for the minimum amount of operating energy. Using this model, a mouldboard plough requiring the least speci"c draught was designed for given soil conditions (speci"c weight of soil, cohesion, soil}metal adhesion, soil}metal frictional angle and soil internal frictional angle) and operating conditions (speed and depth of cut) for a given power availability. Five parameters were set to describe a mouldboard plough, namely width of cut, share angle, side rake angle, angular acceleration of soil inversion in the transverse plane and length of plough in the direction of travel. The designed plough and a commercial mouldboard plough were tested in two di!erent soils in a laboratory soil bin to validate the model and to compare performance. Speci"c draught requirements were compared and it was found that the di!erence between predicted and measured speci"c draught was not signi"cant. The new plough, however, needed signi"cantly less speci"c draught than the commercial plough at identical operating conditions. 2001 Silsoe Research Institute
1. Introduction In agricultural practice, primary tillage is considered as the largest power consuming operation. Many researchers have tried to optimize the performance of mouldboard ploughs in many di!erent ways, mainly with trial and error or semi-theoretical approaches for speci"c operating conditions. For the design of an energy e$cient mouldboard plough in di!erent operating conditions, an understanding of the interactive e!ects of di!erent plough, soil and operational parameters is essential. Optimizing a plough for every soil condition in this manner is cumbersome and almost an impossible task as there may be any number of di!erent conceivable "eld conditions. This made designing a mouldboard plough to suit low-power animal traction a di$cult task. It may be one of the causes of the mouldboard plough not being popular on animal-powered farms. A rigorous mathematical expression is needed to describe the interaction of the factors involved in order to understand the e!ect of di!erent parameters and to optimize the overall performance in di!erent working conditions. 0021-8634/01/040377#13 $35.00/0
Larson et al. (1968) determined the draught force required for a mouldboard plough from a model study, but Wang and Lo (1973) stated that in agricultural tillage work, the economic advantage of small-scale models could be limited. Qiong et al. (1986) presented a model to predict the forces needed to move soil over the mouldboard surface. They concentrated their research on the mouldboard part only and they could predict draught force of an existing mouldboard within 10% of error and side force within 18% of error. Richey et al. (1989) and Ravonison and Destain (1994) used the Bezier bicubic parametric surface model to express the mathematical equation of an existing mouldboard plough. Richey et al. (1989) also tried to predict the draught force, vertical force and side force based on the work of Gill and Vandenberg (1968). The model overestimated draught force by 15%, side force by 53% and vertical force by 115%. Kermis (1978) presented a theoretical analysis of ploughing speed on the turning of the furrow slice by a helical mouldboard plough. However, he did not include the curvature of the plough in the transverse direction but
377
2001 Silsoe Research Institute
378
D . S. S H R ES TH A E¹ A¸.
Notation A
@
A J A N a+ V W X, a+ V W X, a+ V W X, b c c ? D D d F f+ (r, t) V W X, g h i, j, k, R , R , n M B M N mR N P P ? P B P D P D P D P D P J P Q P T R R ,R A A RQ
A
area of plough base resting on the soil, m area of landside resting on the soil, m area of mouldboard, m acceleration in +X, Y, Z, directions, m s\ acceleration in +X, Y, Z, directions in the "rst phase, m s\ acceleration in +X, Y, Z, directions in the second phase, m s\ width of ploughing, m cohesion force per unit area, N m\ adhesion force per unit area, N m\ speci"c draught, N m\ depth of ploughing, m horizontal force required to shear the soil, N +X, Y, Z, co-ordinate function on r and t parameters acceleration due to gravity, m s\ height of soil lifted for inversion, m unit vectors measured speci"c draught, N m\ mass of the plough body, kg mass of soil lifted per second, kg s\ normal force, N total power for ploughing, W power for overcoming adhesive resistance in landside and beneath the plough, W predicted speci"c draught, kPa total power for overcoming friction and adhesion, W power for overcoming friction and adhesion over mouldboard, W power for overcoming friction underneath the plough, W power for overcoming friction on the landside, W power for lifting, W power to shear the soil in front of plough, W power to accelerate the soil, W position vector of any point on plough surface, radius of curvature of plough in the "rst and second phases, m "rst derivative of R , m s\ A
r parameter measuring along the width of plough, m rR , r( "rst and second derivatives of r/R 0A 0A A r width of plough not involved in turning in reference surface, m s a number to distinguish opening furrow with succeeding furrows, ¹ , ¹ , ¹ forces developed by soil acceleration, N t parameter measuring time after soil "rst comes in contact with plough, s t time at end of "rst phase, s < "nal relative velocity of soil in the V X direction with respect to plough, m s\ v speed of ploughing, m s\ v+ velocity in +X, Y, Z, directions at any V W X time in the "rst phase, m s\ v+ velocity in +X, Y, Z, directions at any V W X time in the second phase, m s\ x, y, z co-ordinate of any point on the mouldboard surface, m +y, z,+ , displacement of the soil in +Y, Z, direc tions in the "rst and second phases, m X distance from the soil centre of gravity A to the centre of curvature, m a angular acceleration of soil turning in transverse plane, rad s\ b angular displacement in transverse plane, rad bQ , b$ "rst and second derivative of b with respect to time b maximum value of b, rad K c unit weight of soil, N m\ d soil metal friction angle, deg j share angle, rad
soil internal frictional angle, deg u angular velocity of soil turning at phase transition, rad s\ s soil failure angle, deg t, t side rake angle at the "rst and second phases, rad *A elemental area on mouldboard surface, m *F driving force for soil element, N *F force preventing down sliding of soil on the mouldboard surface, N *m mass of soil column over elemental area *A, kg
D ES I GNI N G A M O U L D BO AR D PLO U G H
assumed it to be straight nor was any share angle considered. Gebresenbet (1995) studied the optimization of animaldrawn curved tillage implements. The study focused on the e!ect of design and operational parameters such as speed, tail angle, side rake angle and radius of a curved soil tillage implement on horizontal and vertical forces. The experiments showed that the e!ect of radius on the horizontal force was di!erent for di!erent tail angles used. However, no great e!ect of radius on the horizontal force was observed. A radius of 270 mm was found to be optimum for a curved implement. For the vertical force, it was observed that for all tail angles used, the force increased with decreasing radius. The objective of the current paper is to optimize the basic design parameters of a conventional mouldboard plough and its operational parameters in relation to its power requirement.
2. Theoretical considerations To simplify the estimation of power required, it is assumed that the total power requirement to pull the plough consists of only the following components: (1) to cut and shear the soil at the leading edge of tool; (2) to overcome adhesion and frictional resistance on the plough surface; (3) to lift the soil for inversion; and (4) to accelerate the soil. It is also advantageous to break up the whole work into components to identify more power-consuming processes.
2.1. Power requirement to cut and shear the soil at the leading edge of the tool For the pure cutting resistance occurring at the leading edge of the tool, Gill and Vandenberg (1968) found that this could be neglected for a sharp cutting edge and homogeneous soil. This paper also neglects pure cutting resistance by assuming a sharp cutting edge; however, depending on plough material and operating condition, the share may wear out quite rapidly. In addition, considering the shallow depth of ploughing in animal-drawn plough, the static earth pressure coming from the sides is also neglected. Figure 1(a) shows the soil failure in front of the share point of mouldboard plough. For the sake of simpli"cation, this soil failure is assumed to be in a straight line and for a shallow ploughing depth, the earth pressure on the sides were neglected. During the "rst opening of the soil there is peripheral resistance on both sides of the
379
Fig. 1. Shear failure of soil in the leading edge of the plough: (a) side view; AD, the plough surface; AB, soil failure line; EB, unploughed section of soil; v, forward velocity of the plough; d, depth of ploughing; v, soil failure angle; (b) free body diagram of failure wedge; c, cohesion; k, unit weight of soil; b, width of cut;
, soil internal frictional angle; N, normal force; F, horizontal force to fail the soil
failure wedge, whereas in the succeeding ploughing after opening, this resistance is only on one side. The free-body diagram of soil wedge is shown in Fig. 1(b). By balancing the forces in vertical and horizontal directions and solving for horizontal force applied F, it can be shown that cbd#bdc/2 tan s F" (sin s#tan cos s) cos s!tan sin s cbd cd # # tan s s tan s
(1)
where: c is the cohesion in N m\; b is the width of ploughing in m; d is the depth of ploughing in m; c is the unit weight of soil in N m\; s is the soil failure angle in deg; is the soil internal frictional angle in deg, and s is a number. The value of s is 1 for the opening furrow and 2 for all succeeding furrows. The last term in Eqn (1) accounts for the peripheral resistance. For simplicity, soil failure angle s can be assumed equal to n/4! /2 which also agrees with the "ndings of Sharma et al. (1994) and others. The power required for failing the soil P is then given by Q P "Fv (2) Q where, v is the speed of ploughing in m s\. This is the minimum amount of power needed to fail the soil. If the plough is not designed properly, there may be heaving of soil in front of the plough due to acceleration of soil. This will greatly increase the draught by putting a surcharge load on the failure wedge.
2.2. Power requirement to overcome adhesion and external frictional resistance In order to determine the adhesion and frictional forces, three surfaces of the plough namely, the landside, the mouldboard surface and the surface in contact between the mouldboard plough base and the ground are considered separately. Figure 2 shows a typical plough base.
380
D . S. S H R ES TH A E¹ A¸.
Fig. 2. Schematic diagram of the modelled mouldboard plough
The power requirement to overcome the adhesive force is directly proportional to the surface area of the plough in contact with the soil. The area of plough in contact with the soil at the base of the plough is dependent on the mouldboard design and the soil type. Theoretically, to minimize the adhesion, only the cutting edge of the mouldboard share and base part of the landside should be resting on the soil. Practically, resting on the soil only with the cutting edge with zero area is impossible; however, for simplicity, this area is neglected and only the basal area of the landside is considered as the area in contact between the plough and the ground. The lateral area of landside coming in contact with the soil is estimated equal to the projected area of the mouldboard surface in the longitudinal}vertical plane up to the depth of ploughing. The power P required to overcome the ? adhesive resistance on the landside and beneath plough is given by P "vc (A #A ) (3) ? ? @ J where: c is the adhesion in N m\; A is the area of the ? @ plough base resting on the soil in m, and A is lateral J area of landside resting on the soil in m as shown in Fig. 2. To determine the power requirement to overcome frictional and adhesive resistance on the mouldboard surface, the entire soil on the plough was divided into elemental blocks of soil as shown in Fig. 3(a). The positive direction of measurement of di!erent axes is shown by the mutually perpendicular unit vectors i, j and k. The origin of this orthogonal co-ordinate system is the intersecting point of the lines drawn, one parallel and another perpendicular to the direction of travel from the share point of the plough as shown in Fig. 3(a) on the XY plane. It has been assumed that the soil mass is a deformable body and can deform while moving along the mouldboard surface. This causes some relative motion between an elemental block of soil to its surrounding soil mass. This introduces the cohesive and frictional resistance in the soil interface. The direction of the cohesive and frictional force depends on the direction of relative soil movement in the vicinity of an elemental block. In this analysis, it has been assumed that the soil is continuously deforming, and hence relative movement of soil on opposite sides of an elemental block is also in opposite directions. So, this force on opposite sides can be assumed equal in magnitude but opposite in direction and they
cancel each other. The driving force *F in Fig. 3(b) is the net e!ective driving force to cause the soil to move along the surface. Let the co-ordinate of any point on the surface of the plough be described by the parametric equation in r and t, r measured along the width of plough, and t measured time after the soil "rst comes in contact with the plough. Plough co-ordinates in the X, Y and Z directions x, y and z are functions of r and t. Mathematically, it can be expressed as: [x, y, z]"[ f (r, t), f (r, t), f (r, t)] (4) V W X The position vector R of any point on the plough surface is given by R"xi#yj#zk
(5)
where, i, j and k are unit vectors in X, Y and Z directions, respectively. A unit directional derivative of position vector R at any point in the r and t directions gives the mutually perpendicular unit vectors on the tangential plane at that point. A cross product of these two vectors generates another unit vector normal to the plane, so there are two co-ordinate systems. The global co-ordinates x, y and z have a "xed origin and direction with respect to the ground, whereas the origin and direction of the local co-ordinate system given by R , R , and n vec tor as shown in Fig. 3(a) changes with position over the plough surface. It should be noticed here that the soil is allowed to slide only along the R direction. Soil movement in the negative R direction is prevented by the soil located below this elemental block. However, the soil element will tend to slip along the slope of the negative R direc tion due to gravity. This downward sliding is prevented by adhesive and frictional resistances. If these resistances are not su$cient to prevent soil from sliding down, a net additional force *F will develop in the soil interface in order to check the sliding, as shown in Fig. 3(c). This means that the resistance force *F may or may not appear depending upon the position of the soil element. As a matter of fact, this force can never be negative in magnitude. If in calculation, a negative value of *F appears, it indicates that the resistance forces are su$cient to prevent downward sliding. Since the soil element can never move in the R direction, the value of *F will be taken as zero. As soil moves only in the R direction, its velocity has to be constant in that direction. If there exists any acceleration, the soil stream will either break apart if it is positive, or will bulk and heave if negative. Also the velocity of the soil element in the direction of n is zero, provided that the magnitude of force supporting the soil N is not negative in Fig. 3(b). Negative value indicates that the soil falls o! before reaching to the end of the
D ES I GNI N G A M O U L D BO AR D PLO U G H
381
Fig. 3. (a) Elemental soil block on plough surface; R1, R2 and n are mutually perpendicular unit vectors in local co-ordinates; i, j and k are unit vectors pointing in X, Y and Z directions; (b) force balanced in the plane perpendicular to R2; DF, driving force; DF1, force preventing down sliding; Dm, mass of soil element; DA, area under soil element; N, normal force; , soil internal frictional angle; d, soil metal frictional angle; c , adhesion; g, acceleration due to gravity; a , a , a , acceleration in X, Y and Z directions; (c) force balanced V W X in the plane perpendicular to R1
plough. Another point to be considered here is that even though soil velocity in the R1 direction is constant, it is not necessary that the R1 vectors for each soil element at a given time along all the length of parameter r are parallel. That is, there may exist some relative velocity in between adjacent elements along the parameter r at a given time. This produces the side frictional force *F tan in the negative R direction as shown in Fig. 3(b). Balancing the forces towards the direction of n in Fig. 3(b) : N!*F tan !*F tan "¹ (6) where: ¹ "*m+(g#a )kn#a in#a jn, (7) X V W in which: ¹ is the component of force in N developed by soil acceleration and dependent on plough geometry; *m is the mass of soil element in kg; g is acceleration due to gravity in m s\; a , a and a are acceleration of soil V W X element in the X, Y and Z directions, respectively, in m s\. At this point, it is unknown whether frictional and adhesive resistance is su$cient to prevent downward
sliding of the soil element. Assuming that the frictional and adhesive resistances are not su$cient and there exists some positive value of *F , then balancing the force in the R direction in Fig. 3(b), it can be shown that !N tan d#*F!*F tan "¹ (8) where: ¹ "*m+(g#a )(kR )#a (iR )#a ( jR ),#C *A X V W ? (9) *A is the elemental area in mouldboard surface in m; and d is the soil metal frictional angle in degree. Again balancing the forces in the direction of R in Fig. 3(c): N tan d#*F tan #*F "¹ (10) where: ¹ "*m+(g#a )(kR )#a (iR ) X V #a ( jR ),!C *A (11) W ? Solving simultaneously Eqns (6), (8) and (10) for N, *F and *F :
¹ (1#tan )#¹ (tan !tan )#¹ (tan #tan ) N" 1#tan #2 tan d tan
(12)
¹ (tan d!tan d tan )#¹ (1#tan d tan )#¹ (tan #tan d tan ) *F" 1#tan #2 tan d tan
(13)
!¹ (tan d#tan d tan )!¹ (tan #tan d tan )#¹ (1!tan d tan ) *F " 1#tan #tan d tan
(14)
382
D . S. S H R ES TH A E¹ A¸.
Fig. 4. Lifting and inversion of soil during ploughing: (a) plough share is advancing through reference plane; MN, share; b width of cut; r width of plough not involved in soil turning; k, share angle; (b) vertical cross-section of soil being lifted in xrst phase; w, side rake angle; R 1, radius of curvature; y1, Y co-ordinate; z1 , Z co-ordinate; d, depth of ploughing; (c) second phase of soil inversion; b, angular A displacement in transverse plane; R 2, radius of curvature; y2, Y co-ordinate; z2, Z co-ordinates; w, side rake angle; (d) end of second A phase; CG, centre of gravity; X , distance of CG from O; b maximum value of b A K
The above assumption that *F is greater than zero is valid only if its value given by Eqn (14) is non-negative. If it is negative, then the value of *F will be taken as zero and Eqn (10) will not be considered. Then substituting *F "0 into Eqns (6) and (8) and solving for N and *F, ¹ #¹ tan
N" (15) 1!tan d tan
¹ tan d#¹ *F" 1!tan d tan
(16)
The total power required for overcoming this frictional resistance over the plough P for the whole mass is given D by P " *Fv (17) D The power required for overcoming friction between the soil and under the plough base is equal to the vertical load on the XY plane coming from plough weight plus dynamic weight of soil which is greater than static load because of the accelerations in the Z direction. The total dynamic load can be determined with reference to Fig. 3(b). Then the frictional force P is given by D P "[M g# +N(nk)#*F(R k)#*F (R k),]v tan d D N (18) where, M is the mass of plough body. Referring to N Fig. 3(c) the power required to overcome the friction on the landside P is given by D P " +N(nj)#*F(R j)#*F (R j),v tan d (19) D The total power P required to overcome the frictional D and adhesive force is the summation of power from Eqns
(3), (17), (18) and (19) which is given by P "P #P #P #P D ? D D D
(20)
2.3. Power requirement for lifting the soil The power requirement for lifting the soil mass is minimum when the soil is lifted only to its minimum required height. Figure 4 shows the cross-sectional view of soil mass from the transverse direction. To prevent the soil from over lifting, the upward velocity of the soil mass centre of gravity (CG) in Fig. 4(d ), should be zero at the time when the soil leaves the plough. In order to invert the soil, the ploughed section ACB will have to rotate about point A at least until it reaches the position shown by Fig. 4(d ). In this position, the CG of the soil section comes directly over point A. The rest of the inversion will take place due to the inertia of the soil mass in the transverse direction. The minimum height of the soil to be lifted is derived in Eqn (49) later. The mass of the soil lifted per second mR is given by mR "bdvc/g
(21)
Therefore, the power required for lifting the soil P is J given by P "mR g(h!d/2) (22) J where h is the height of the CG as shown in Fig. 4(d ) in m. 2.4. Power requirement to accelerate the soil To determine the power required to accelerate the soil, only the initial and "nal conditions of the soil blocks are
383
D ES I GNI N G A M O U L D BO AR D PLO U G H
considered. At the beginning, when the soil "rst comes in contact with the plough, its relative velocity is equal to the speed of ploughing. As the plough advances, the soil changes direction but the plough does not, and hence their relative velocity changes. The energy imparted to the soil mass per second will be equal to the power requirement to accelerate the soil P , which after simpli"T cation can be shown as P "bdcv(v!< )/g (23) T V where < is the relative velocity of soil in the direction of V travel when it leaves the plough. So, the total power required in ploughing P is given by P"P #P #P #P Q D J T And the speci"c draught force is given by P D" D vbd
(24)
considerations for the two processes. The following sections consider these phases in detail. 2.5.1. First phase In Fig. 4(b), the plough is moving away from the viewer perpendicular to the paper. When soil slice starts bending, the curved length of each layer remains essentially unchanged. From Fig. 4(b), the width of the plough r that is not involved in turning the reference surface is given by r "b!vt tan j
(26)
where t is the time period in s after the soil furit comes in contact with the plough. It is assumed that the side rake angle t is constant for a given plough until the end of the share (point N) passes the reference plane. If this angle is expressed in radians, the radius of curvature R is given A by
(25)
2.5. Mathematical expression for soil movement along the mouldboard surface The following assumptions are made while deriving mathematical expressions for soil movement along the mouldboard surface: (1) the soil is homogeneous and isotropic; (2) the bulk density of the soil remains unchanged during movement over the plough surface; (3) the soil mass does not fall o! before it reaches the end of the plough (where it is intended) and the normal force exerted by the plough on the soil is not negative; (4) the soil can break up along the normal direction of the plough surface but the adjacent surface remains in contact; (5) the side rake angle t of the plough is constant until all the length of the share in the direction of travel passes through a reference surface (as shown in Fig. 4); and (6) the share angle j of a plough is constant. During the ploughing process, "rst of all the tip of the share (point M) comes in contact with a reference surface, which is in the vertical plane, as shown in Fig. 4(a). The time t required for all the share length to pass the reference surface depends upon the width of the plough b, forward speed of plough v and the share angle j. The turning process of soil before and after time t is di!erent. Before time t [the "rst phase of turning as shown in Fig. 4(b) ], the entire width of the plough is not involved in turning the soil, whereas after t [the second phase of turning as shown in Fig. 4(c) ] all the width of the plough is involved in turning the soil. This necessitates separate
b!r R " A t
(27)
Let r be any length measured from point A along the width of the plough and varied from zero to b. The vertical lift z of any arbitrary point C at distance r, such that r*r , is given by
z "
b!r t
1!cos
(r!r ) t b!r
(28)
In the Y direction, the co-ordinate of the same point y , following the direction convention shown in Fig. 3(a), is given by
y "!
b!r (r!r ) t sin #r b!r t
(29)
When t"0, r"r "b and Eqns (28) and (29) become indeterminate. This is plausible, as at zero time the radius of curvature and arc length both are zero. This situation was solved by taking the limiting value. That is as tP0, r Pb, the value of z becomes zero and y becomes r . The velocity in the Z and Y directions v and v can be X W obtained by di!erentiating Eqns (28) and (29) with respect to time as b!r (r!r )t v " v tan j sin X b!r (b!r )
v tan j (r!r )t # 1!cos t b!r
(30)
b!r (r!r )t v tan j cos v "v tan j! W b!r (b!r ) v tan j (r!r )t ! sin t b!r
(31)
384
D . S. S H R ES TH A E¹ A¸.
These equations are also evaluated by taking the limiting value at t"0. The velocity in the X direction can be determined using the following equation: v "(v!v !v (32) V X W Accelerations in Z and Y directions a and a can be X W obtained by di!erentiating Eqns (30) and (31) with respect to time, as
where b is the angular displacement at any time after t , and given by b"u (t!t )#1/2a (t!t ) (40) where u is the angular velocity of rotation in rad s\ at t"t , and a is the angular acceleration in rad s\. Vel ocities in Z and Y direction are given by di!erentiating Eqns (38) and (39) with respect to time as
(b!r)t (r!b#vt tan j)t a " cos X vt tan j vt tan j
(33)
(r!b#vt tan j)t (b!r) t sin a " W vt tan j vt tan j
(34)
RQ z r v " A #R !bQ sin b#(bQ #rR ) sin b# X A 0A R R A A (41)
These equations are also evaluated taking limiting values at t"0. The acceleration in the X direction can be obtained by di!erentiating Eqn (32) and is given by
RQ y r v " A !R !bQ cos b#(bQ #rR ) cos b# W A 0A R R A A (42)
v a #v a W W a "! X X V v V
(35)
2.5.2. Second phase After the end of the "rst phase at time t , the soil mass is rotated about point A as shown in Fig. 4(c). To avoid the instantaneous change in velocity at the transition between the two phases, the radius of curvature of the plough has to increase at the same rate as before. If the radius of curvature is kept constant after transition, the velocity of the soil elements at the end of phase 1 will not be equal to the velocity of the same soil element at the beginning of phase 2 for all radii at any positive value of side rake angle t. This is because, during phase 1, the radius of rotation of soil elements with reference to point A is continuously decreasing at di!erent rates for di!erent points on the plough. Hence, for a smooth transition this decreasing radius should be kept the same as in phase 1. Combining Eqns (26) and (27), and replacing R by R , radius of curvature at any time is given by A A vt tan j R " (36) A t Since the width of plough b is constant, this increase in radius of curvature will decrease the side rake angle t. Let this new side rake angle be t at any time t, then bt t " vt tan j
(37)
z and y co-ordinates of any point z and y at distance r as shown in Fig. 4(c), are given by
r z "R cos b!cos b# A R A r y "!R sin b# !sin b A R A
(38) (39)
where bQ , RQ and rR are the "rst derivatives of b, R , A 0A A and r/R , respectively, with respect to time. At the A transition between the two phases, the velocity in both Y and Z directions given by Eqns (30), (31) and (41), (42) must be equal. The value of t can be obtained from Eqn (26) after substituting r "0. Substituting this value of t in these equations, the angular velocity at time t can be solved as tv tan j u " b
(43)
Before the soil leaves the plough, the soil centroid will be at least above point A as shown in Fig. 4(d). After this position, the soil will fall to the right side because of the moment of inertia in that direction. When the radius of curvature R is large in comparison to the depth and A width of the plough, the CG of the soil mass will be closer to the point E, which is the midpoint of the centre line. But, whether the CG will be on the right or left side of point E depends on the thickness of soil and the subtended angle t . The distance X from the CG to the A centre of curvature is given by 4 R !(R !d) sin(t /2) A (44) X " A A 3 R !(R !d) t A A When the CG is exactly above point A, the horizontal components of X and R will be equal. That is, A A t X sin b # "R sin b (45) A K A K 2
where, b is the maximum value of b. After substituting K values for R , t b and X from Eqns (36), (37), (40) and A A (44) into Eqn (45), the maximum value of t can be solved by trial and error method. Then, the velocity in the X direction is given by: v "(v!v !v V X W
(46)
385
D ES I GNI N G A M O U L D BO AR D PLO U G H
The length of the plough in the direction of travel was calculated by integrating Eqns (32) and (46) with respect to time numerically for their corresponding interval. So, this length of the plough is the dependent parameter describing a plough and can be calculated knowing other parameters at optimum condition. From Fig. 4(d ), the value of h is given by
t h"R cos b #X cos b # K A K A 2
(47)
This equation can be substituted into Eqn (22) to determine the power required for lifting the soil mass on the plough base. After di!erentiating Eqns (41) and (42), accelerations in the corresponding directions are given by v R !RQ z X A A a "2RQ X A R A !(bQ ) cos b!b$ sin b#+bQ #rR #R
A
, 0A
r r cos b# #+b$ #rK , sin b# 0A R R A A
(48)
RQ y !v R A W A a "2RQ W A R A (bQ ) sin b!b$ sin b!+bQ #r , 0A !R r r A sin b# #+b$ #rK , cos b# 0A R R A A
(49)
where b$ and r( are the second-order derivatives of 0A b and r/R with respect to time. These values of accelerA ation, velocities and displacements are used to estimate the power required in di!erent components as discussed earlier. Acceleration in the X direction a can be calV culated di!erentiating Eqn (46) and is similar to Eqn (35) after substituting subscript 2 for 1.
3. Experimental details For design purposes, the draught requirement to open the furrow has been considered. The plough surface was divided into small rectangular sections, 40 sections wide by 57 sections long. The optimum values of parameters were found by trial and error using the derived equations and Matlab software. The plough was designed for the soil, operating and other parameters shown in Tables 1}3. These physical parameters were measured at the workable conditions of the soil in a soil bin. Soil to metal adhesion and soil metal frictional angles were measured
Table 1 Measured soil parameters for designing a mouldboard plough Soil parameters
Value
Moisture content,% db Cohesion, N m\ Internal frictional angle, deg Wet speci"c weight, kN m\ Adhesion, N m\ Soil}metal frictional angle, deg
24)28 8300 11)2 18)1 6400 8)01
at a spead of 0)4 m s\, whereas cohesion was soil internal frictional angle were measured in a quasi-static condition using the torsional shear apparatus. Two types of soil were selected to test the plough (Table 4). All "ve soil parameters were measured at the lower moisture range. Then after carrying out a set of experiments with both the new and commercial plough, some water was added to increase moisture content. The soil was again mixed thoroughly and levelled to take the next set of measurements. This procedure was repeated for both soils to make three di!erent moisture conditions in each soil. Moisture content of silty clay loam was 28)8, 31)2 and 34)4% and for clay it was 40)4, 41)1 and 47% dry basis, on average. The ploughing speed varied between 0)3 and 0)4 m s\. For each ploughing speed, the depth of ploughing was varied from 8 to 12 cm at intervals of 2 cm. A commercial mouldboard plough comparable in size to the modelled plough was bought as shown in Fig. 5. The detailed dimension of the commercial plough was not available and there was not any de"nite speci"cation. Both the modelled plough and a comparable commercial plough were tested in the same soil and operating conditions. In the three soil conditions in each type of soil, three opening furrows were made by the model plough and the other three by the commercial plough in such a way that each plough made at least one opening furrow in each soil. For the opening furrow, only the depth was varied, and the speed kept constant at 0)4 m s\. In this way, a total of 45 di!erent test runs were made for each plough. Measured values of draught at di!erent conditions were then compared with calculated values of
Table 2 Assumed operating conditions for optimization Ploughing parameters Depth of ploughing, m Speed of ploughing, m s\ Power available, W
Value 0)10 0)40 300}400
386
D . S. S H R ES TH A E¹ A¸.
Table 3 Other necessary parameters involved in designing the plough Other parameters
Value
Steel density, kg m\ Acceleration due to gravity, m s\ Thickness of mouldboard, m Thickness of landside, m
7835)6 9)81 0)004 0)01
Table 5 Optimum plough parameters for the given soil and other parameters Design parameters
draught. The speci"c draught required for each operating condition for modelled and commercial ploughs were computed and statistical operations was performed to analyse the results.
4. Results and discussion While studying the theoretical e!ect of one parameter, all other parameters were kept constant at the design values given in Tables 2, 3 and 5.
4.1. E+ect of soil parameters The analysis showed that the e!ect of cohesion on power required to fail the soil in shear was essentially a linear relationship. When cohesion varied from 0 to 15 kPa, power required to shear-fail increased from 10 to 240 W, and the speci"c draught increased from 28 to 66 kPa. The e!ect of internal frictional angle on the power for adhesive and frictional resistance was rather linear. However, up to a value for the internal fritional angle of 103, the value of normal force, N given by Eqn (12) was negative in some parts of the rear section of the plough. This indicates that some soil falls o! before reaching the end of the plough, and hence, the predicted speci"c draught for values of the internal frictional angle below 100 was non-linear. However, in practice in normal "eld conditions, this angle is almost always above 103. The power required for lifting and accelerating the soil was
Value
Width of cut, m Share angle, deg Side rake angle, deg Angular acceleration, deg/s Calculated power requirement, W
0)15 41)5 90 2)3 326
low at tested range of speed. When this angle was increased from 10 to 403, the speci"c draught increased from 54 to 81 kPa. The power to fail the soil increased from 165 to 310 W although the relationship was nonlinear. The power to overcome friction and adhesion increased from 130 to 150 W, and power for accelerating and lifting remained unchanged at 20 and 4 W, respectively. At the lower range of adhesion, the power required to fail the soil was greater than the power required to overcome friction and adhesion, whereas at higher condition the power required to overcome friction and adhesion was greater. The value of power required for lifting and accelerating the soil was rather small, and independent of adhesion. The soil metal friction angle had no e!ect on the power to fail, lift and accelerate the soil. The power to overcome frictional and adhesive resistance increased with increasing soil to metal friction angle. When this angle was increased from 10 to 403, the power required to overcome friction increased from 130 to 220 W. The power required to fail the soil and to overcome adhesion and friction at normal operating conditions required about 92% of the total power, whereas accelerating and lifting needed the remaining 8%. The speci"c weight of the soil varied from 12 to 24 kN m\. In this range, the speci"c draught increased from 52 to 57 kPa, the power required to fail the soil increased from 67 to 172 W, the power to overcome friction and adhesion increased from 129 to 137 W, the
Table 4 Particle size distribution of test soil Particle size distribution, % Text soil
Soil type
Bin 1 Bin 2
Silty clay loam Clay
Sand ('0)075 mm)
Silt (0)075}0)002 mm)
Clay ((0)0002 mm)
Specixc gravity
11)1 17)2
59)1 28)9
29)8 53)9
2)44 2)51
D ES I GNI N G A M O U L D BO AR D PLO U G H
387
Fig. 5. (a) Model plough with mouldboard extension; and (b) commercial plough
power to accelerate the soil increased from 13 to 27 W and the power to lift the soil increased from 3 to 5 W. For the observed range of speci"c weight, the variation in power was small and essentially linear in character.
4.2. E+ect of operating parameters The results showed that if the depth of ploughing is less than the design depth, the designed plough was unable to completely turn the soil, as shown in Fig. 6. This e!ect can be explained from Fig. 4(d ). As the thickness of the soil slice reduced, for the plough position shown, the CG of the soil will still be on the left side of the vertical plane through point A, and hence the soil will be left unturned. However, placing a mouldboard extension to further turn the soil as in Fig. 5(a) improved soil inversion. Soil inversion was better when the ploughing depth was equal to the design depth of 10 cm (Fig. 7). If the depth of the cut is greater than the width of the cut, during inversion, the "nal position of the CG of the soil mass will be in a lower position than the original and soil need not be lifted. Instead some of the potential energy of the soil will be utilized to move the soil mass and reduce the power requirement to accelerate the soil. But as depth increases, the turning of soil on the "rst pass was di$cult, even after putting on a mouldboard extension. Lab-
Fig. 6. Partial inversion of soil while ploughing at lower than design depth (8 cm depth of ploughing)
oratory tests have shown that soil fell back into the furrow as depth increased as shown in Fig. 8. The inversion in the case of the commercial plough was not continuous. For the observed set of tests, it carried some soil along with the plough as can be seen in Fig. 8. This may be one of the reasons that this implement needed relatively higher speci"c draught. At higher speeds than the designed speed, the velocity of the soil mass moving over the plough in the X direction in some elemental area given by Eqn (46) became imaginary. This, however, is not a valid condition. When the plough was operated in the soil bin at higher speed, instead of inverting and throwing soil, the plough carried soil along with it, which resulted in bulking and heaving of soil on the plough base as shown in Fig. 9. When the speed was increased from 0)3 to 0)4 m s\, the speci"c draught requirement increased from 33 to 54 kPa. The power to shear-fail increased from 130 to 170 W, power to overcome friction and adhesion increased from 60 to 130 W, the power to accelerate increased from 5 to 20 W and the power to lift the soil increased from 1 to 4 W.
4.3. Experimental results From the mechanical and hydrometer analyses, the proportions of sand, silt and clay were determined and
Fig. 7. Furrow made at a design depth of 10 cm
388
D . S. S H R ES TH A E¹ A¸.
Fig. 8. Some soil fell back into the furrow at 12 cm depth of ploughing
are shown in Table 5. The calibration curve for the load cell and octagonal ring transducer produced a linear relationship with coe$cient of determination R of at least 0)999. There was no measurable amount of hysteresis e!ect, nor cross sensitivity due to forces in directions other than the measured direction. The measured and predicted draught for the modelled plough for both soils in various operating conditions were plotted in a scatter plot as shown in Fig. 10. To con"rm the correlation between measured and predicted values of draught, a paired sample t-test was performed. A null hypothesis was set, that there was no signi"cant di!erence between predicted and measured values, against the alternative hypothesis that there is a signi"cant di!erence between measured and predicted speci"c draught. A two-tailed test at the 95% level of con"dence showed that there was no signi"cant di!erence between measured and predicted values of speci"c draught. From the statistical analysis, the following equation of best-"t line was obtained:
Fig. 10. Scatter plot for predicted and observed specixc draught for modelled mouldboard plough at diwerent operating conditions
above regression line was found to be 1)2$0)22 and 5)4$7)4, respectively. A scatter plot of measured speci"c draught of the modelled and commercial ploughs is shown in Fig. 11. Only the result for succeeding furrows could be compared as only one plough was tested in one soil condition as the opening furrow. A paired one-tailed t-test, with a null hypothesis that speci"c draught for both ploughs are not signi"cantly di!erent against an alternate hypothesis that speci"c draught for the model plough is signi"cantly lower than commercial plough, was carried out. The result showed that speci"c draught required for the model plough is signi"cantly lower than the speci"c
M "1)2P !5)4 (50) B B where M and P are measured and predicted speci"c B B draughts, respectively, in kPa. A statistical analysis at the 95% con"dence interval of slope and intercept of the
Fig. 9. Heaving of soil at high speed
Fig. 11. Scatter plot for measured specixc draught for model and commercial plough at diwerent operating conditions
D ES I GNI N G A M O U L D BO AR D PLO U G H
draught of the commercial plough. On average the draught required for modelled plough was 10)6$5)6% (with 95% con"dence interval) less than the commercial plough.
5. Conclusion A mathematical model of a mouldboard plough was developed and the equations describing the draught force requirement were evaluated and iterated to get the optimum value of parameters of mouldboard plough. Five soil parameters were considered to a!ect the draught force requirement: cohesion, soil internal frictional angle, soil}metal adhesion, soil}metal friction angle and speci"c weight of the soil. The two operating parameters considered were depth of ploughing and speed of ploughing. Based on these seven parameters, four plough parameters were optimized in order to minimize the speci"c draught requirement: width of cut, share angle, side rake angle and angular acceleration during soil inversion in transverse plane. Power availability and speed of ploughing were the major factors determining the overall size of the plough. Analyses showed that the main power requirements of the ploughing process were failing the soil in shear and overcoming adhesion and frictional resistance on the plough surfaces. These two components of power accounted for about 92% of total power in the normal conditions considered in this research. The power requirement for lifting the soil was of least magnitude among all four components of power considered. Statistical analyses of the soil bin test results of the model plough showed that there was no signi"cant di!erence between measured and predicted speci"c draught. On average speci"c draught required for the model
389
mouldboard plough was 10)6$5)6% less than commercial mouldboard plough. It was observed that the modelled plough could not invert the soil properly when operating at lower than design depth. However, adding a mouldboard extension improved the inversion. Performance comparison of these two ploughs showed that the soil inversion was better in case of the model plough than the commercial plough.
References Gebresenbet G (1995). Optimization of animal drawn tillage implements, part I: performance of a curved tillage implement. Journal of Agricultural Engineering Research, 62(3), 173}184 Gill W R; Vandenberg G E (1968). Soil dynamics in tillage and traction. Agricultural Handbook No. 316. Agricultural Research Service, United States Department of Agriculture Kermis L H (1978). A theoretical approach to high speed plough design. Journal of Agricultural Engineering Research, 23(4), 343}368 Larson L W; Lovely WG; Bockhop C W (1968). Predicting draft forces using model moldboard plows in agricultural soils. Transactions of the ASAE, 11(5), 665}668 Qiong G; Pitt R E; Ruina A A (1986). Model to predict soil forces on the plough mouldboard. Journal of Agricultural Engineering Research, 35(3), 141}155 Ravonison N M; Destain M F (1994). Parametric cubic equation for modelling moldboard plow surfaces. Soil and Tillage Research, 31(4), 363}373 Richey S B; Srivastava A K; Segerlind L J (1989). The use of three dimensional computer graphics to design mouldboard plough surface. Journal of Agricultural Engineering Research, 43(3), 245}258 Sharma V K; Singh G; Gee-clough D (1994). Force}time behaviour of #at tines in dry sand. Journal of Agricultural Engineering Research, 57(3), 191}197 Wang J K; Lo K (1973). Predicting moldboard plow draft in di!erent processed soils. Transactions of the ASAE, 16(5), 851}854
